However, the counterfactual fairness is a causal quantity that is not necessarily unidentifiable from observational data without the knowledge of structure equations [1]}. In [2]}, the authors studied the criterion of identification of counterfactual fairness given a causal graph and provided the bounds for unidentifiable counterfactual fairness.
According to Proposition 1 in [2]}, our counterfactual fairness is identifiable if \(\mathbf {X}\backslash \lbrace S\rbrace  \)  are not descendants of \(S\) . In this case,
similar to Eq. (REF ), we have that \(\mathbb {E}[R(\pi , s^*)| \mathbf {x}_t ] = \mathbb {E}_{\mathbf {a}\sim \pi } \left[ \mathbb {E}[R(\mathbf {a},s^{*})|\mathbf {x}_{t}] \right]\)  where \(s^* \in \lbrace s^+, s^-\rbrace \) . Similar to Eq. (REF ), we denote \(\mu _{a,s^*} = \mathbb {E}[R(a,s^*)|\mathbf {x}_{t}]\) , which can be decomposed using the do-calculus as
\(\mu _{a,s^*} = \mathbb {E}[R(a,s^*)|\mathbf {x}_{t}] = \sum _{\mathbf {Z}} \mathbb {E}[R| s^{*},\mathbf {w}\backslash s_t] \cdot P(\mathbf {z}|s^*, \mathbf {x}_{t,a} \backslash s_t)\) 
